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Basic Mathematics

forAstronomy

A Manual for Brushing off the Rust

by

Dr. Glenn Tiede2007

Version 1.2 August 10, 2007


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Introduction

Astronomy is a fascinating science, from the distances to and inter-workings ofstars and planets to the big questions like, Where did it all come from?, What isultimately going to happen? and Are we alone?. Before we can begin to investigate

these concepts we first have to master some basic mathematical skills that will allow us totalk about these astronomical concepts.

Most of these skills are grade school level arithmetic skills, so you dont have tobe a mathematics major to understand the mathematical concepts necessary to do well inan introductory astronomy class. Honestly, you dont even have to like math. What youdo have to do is be willing to spend the time necessary to master these skills. In somecases you may already have the mastery. In others, it may just be a matter of a littlepractice to knock the rust off. Whatever the case, just like professors in other subjectscant take time from class to remind you how to read or write, we wont be spending anytime in our astronomy class reviewing basic mathematical skills. Therefore this manual

provides you a quick guide to the mathematical skills you will need and a basic set ofpractice exercises so you can check your level of mastery.

Each section in this manual begins with a description of each set of skills. Youshould read these descriptions and if you fully understand the concepts, try some of thesample problems. If you score well, move on to the next section. However, if you findthat you do need to rethink these concepts, then study the description carefully and do allof the exercises section. The questions in the exercises section go more in depth thanmultiple choice exam questions will, but doing them will help you think through andmaster the concepts. Answers for all of the exercise questions are provided in the lastsection. Only use these answers after you are done with the exercise questions (looking at

the answers ahead of time will not help you learn). Finally, the last part of each sectionis example exam questions. These are the types of questions you can expect to find onmultiple choice exams related to these skills.

If at any point you are having difficulty with any of the sections, there are manyfurther resources you can use. First, as you get to know people in the class you can formstudy groups. Study groups are excellent not only to work through this booklet, but tostudy for the class in general. Secondly, the web provides many sources of basicmathematical information. Try googling the term for whatever point you are stuck on.If this doesnt work, the University provides an entire academic unit, the Mathematicsand Statistics Tutoring Center, which is there to help you with mathematical concepts.They are located at 208 Moseley Hall, (419) 372-8009, and provide services free ofcharge. Finally, your professor is there to help you as well. If its a quick question askbefore or after class, or send email. If you need to discuss a point in more depth, drop byoffice hours.

Remember mathematics, like all of the other skills you are gathering here at theUniversity, is a tool you need to master to help you succeed. If your earlier mathematicstraining was lacking due to poor curricula or poor teachers, your college career is yourlast chance to master these very important skills.


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Names of Numbers

Description

One of the first mathematical challenges we find ourselves facing in astronomy is

dealing with very large numbers. Unless you are an accountant for the federalgovernment, these are numbers you just dont encounter in everyday life. Before we canbegin to even talk about them we need words to name these very large numbers.

Numbers are named following a very straightforward pattern. Starting with thefamiliar, one thousand, and progressing to larger numbers it goes as follows:1

1,000 one thousand1,000,000 one million

1,000,000,000 one billion1,000,000,000,000 one trillion

Notice how each set of three zeros (separated by a comma) has a new name. Note theorder of the names, thousand, million, billion, trillion, each greater than the previous.The above are the names you need to be familiar with for you introductory astronomyclass, but in case you are interested, the names continue.

1,000,000,000,000,000 one quadrillion1,000,000,000,000,000,000 one quintillion

1,000,000,000,000,000,000,000 one sextillion1,000,000,000,000,000,000,000,000 one septillion

1,000,000,000,000,000,000,000,000,000 one octillion

1,000,000,000,000,000,000,000,000,000,000 one nonillion1,000,000,000,000,000,000,000,000,000,000,000 one decillion... etc.

Each name represents a number one thousand times larger than the previous, and since wecan always multiply any number by 1,000 there is no largest number, and the namingconvention continues. But concentrate on the first table, thousand, million, billion, andtrillion. If you can keep these straight, which is larger than which, and how many digitseach has, you are set for introductory astronomy.

Now of course there are numbers in between one million and one billion, onebillion and one trillion and we need names for these as well. Fortunately we dont haveto memorize billions of names since all of these numbers follow a very straight forwardnaming convention, and if you can count to one thousand, you already know it. Lets startwith million.

1 The British and some other Commonwealth countries use a different naming progression. The onepresented here is the American system which is becoming the most commonly used, even in England.


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1,000,000 one million2,000,000 two million3,000,000 three million5,000,000 five million

10,000,000 ten million

20,000,000 twenty million50,000,000 fifty million100,000,000 one hundred million250,000,000 two hundred fifty million585,000,000 five hundred eighty-five million

Notice that the six zeros to the right are just there to tell us million and the digits left ofthese zeros are telling us how many million. This pattern repeats for billion, trillion, andall numbers larger. For example:

2,000,000,000 two billion10,000,000,000 ten billion13,000,000,000 thirteen billion

151,000,000,000 one hundred fifty-one billion2,000,000,000,000 two trillion8,000,000,000,000 eight trillion

214,000,000,000,000 two hundred fourteen trillion

Finally, we can mix these all together. For example:

1,854,000 one million, eight hundred fifty-four thousand

12,130,000 twelve million, one hundred thirty thousand2,064,600,000 two billion, sixty-four million, six hundred thousand

8,284,347,418,671 eight trillion, two-hundred eight-four billion, threehundred forty-seven million, four hundred eighteenthousand, six hundred seventy one

With the numbers in this last set, especially the very last one, you can already seen one ofthe advantages mathematics has as a tool. It is much easier to write the number withnumerical digits than it is to write it out in English.


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Exercises

A) Write the following numbers numerically.

1) one million ______________________________________

2) two billion ______________________________________

3) five trillion ______________________________________

4) twenty-three million ______________________________________

5) fifty-seven trillion ______________________________________

6) two hundred thirty-eight billion ______________________________________

7) seventy-eight billion,three hundred eighty-eight million ______________________________________

8) four hundred fifty-three billion,eight hundred sixty-one million ______________________________________

9) five hundred sixty-nine trillion,four hundred seventy-one billion,two hundred thirty-two million ______________________________________

10) nine hundred fourth-seven trillion,six hundred ninety-two billion,twenty-one million,six hundred one ______________________________________

B) Write the names of the following numbers in English.

1) 3,000,000 ______________________________________

2) 5,000,000,000 ______________________________________

3) 7,000,000,000,000 ______________________________________

4) 56,000,000 ______________________________________

5) 73,000,000,000,000 ______________________________________

6) 868,000,000,000 ______________________________________

7) 97,142,000,000 ______________________________________

______________________________________

8) 885,479,000,000 ______________________________________

______________________________________

9) 421,933,593,000,000 ______________________________________

______________________________________

______________________________________


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10) 237,807,046,612,739 ______________________________________

______________________________________

______________________________________

______________________________________

______________________________________

Sample Exam Questions

1) Which of the following is the numerical equivalent of five million?a) 5,000,000b) 50,000c) 500,000d) 5,000,000,000,000

2) Which of the following is the numerical equivalent of eight trillion?a) 8,000,000,000,000,000,000,000b) 8,000,000c) 8,000,000,000,000d) 800,000,000

3) Which of the following is the is the name for 500,000,000?a) five hundred thousand millionb) five hundred trillionc) five hundred billiond) five hundred million

4) Which of the following is the name for 434,346,551,000?a) four hundred thirty-four trillion, three hundred forty-six billion, five

hundred fifty-one thousandb) four hundred thirty-four billion, three hundred forty-six million, five

hundred fifty-one thousandc) four hundred thirty four kzillion, three hundred forty-six bzillion, five

hundred fifty-one million billiond) four hundred thirty-four million, three hundred forty-six thousand, five

hundred fifty-one

5) Of the following four numbers, which is largest?a) 900,342,797b) one hundred three trillionc) 141,075d) two hundred ninety-two billion


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Metric Prefixes & Abbreviations

While the naming convention for numbers is straightforward, writing the namesout in English is rather cumbersome. Further, million, billion, and trillion sound verysimilar and can be confused when spoken. Finally, these are English words. People who

dont speak English will not understand them. For all of these reasons when the MetricSystem was formally defined a set of prefixes for units and abbreviations for theseprefixes were also defined. In 1991 the 19th General Conference on Weights andMeasures extended the list of prefixes so that it encompasses numbers both small andlarge enough that they span all of the uses of modern science and technology. Someprefixes, kilo- (K), mega- (M), and giga- (G) have entered everyday language and areexamples of ones we will most often encounter in astronomy. The complete set is:

Prefix Abbreviation Number Name Number in Scientific

Notation2

yotta- Y 1 septillion 1024

zetta- Z 1 sextillion 1021

exa- E 1 quintillion 1018

peta- P 1 quadrillion 10 15

tera- T 1 trillion 1012

giga- G 1 billion 109

mega- M 1 million 106

kilo- K or k 1 thousand 103

hecto- h 1 hundred 102

deca- da ten 101

deci- d 1 tenth 10-1

centi- c 1 hundredth 10-2

milli- m 1 thousandth 10-3

micro- 3 1 millionth 10-6

nano- n 1 billionth 10-9

pico- p 1 trillionth 10 -12

femto- f 1 quadrillionth 10-15

atto- a 1 quintillionth 10-18

zepto- z 1 sextillionth 10-21

yocto- y 1 septillionth 10-24

The prefixes highlighted with a light gray background are the ones you will encounterregularly in introductory astronomy.

Metric prefixes are not used in place of numbers themselves but rather with unitsof measure. For example, you would not replace billion in a sentence with giga, butyou can write one billion years as one giga-year or more simply and compactly, 1 Gyr.

2 The following section explains scientific notation.3 This is the Greek letter mu, the Greek lowercase m. It is the only non-roman letter used as a metric

prefix. Greek letters and why we use them in astronomy are detailed in a later section.


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Exercises

A) Write the following quantities with their metric prefixes. Use the followingabbreviations for units:

seconds syears yr meters m

1) 1 million seconds _____________________________

2) 3 billion years _____________________________

3) 700 billionths of a meter _____________________________

4) 1000 meters _____________________________

5) 1,000,000 kilometers (in units of meters) _____________________________

B) Write out the following abbreviated measures in English:

1) 8 Ms _____________________________

2) 4.5 Gyr _____________________________

3) 300 nm _____________________________

4) 8 m _____________________________

5) 1 cm _____________________________


1) One billion, 500 million seconds is abbreviated _____.a) 1.5 Msb) 15 Msc) 15 Gsd) 1.5 Gs

2) 4 billion 650 million years is abbreviated _____.a) 4.65 gyr

b) 4.65 Gyr c) 4.65 cyr d) 4 Gyr, 6.5 cMyr


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3) How is 550 nm written out in English?a) Five hundred fifty million meters.b) Fifty-five hundred billionths of a meterc) Five hundred fifty billionths of a meterd) Five hundred fifty trillionths of a meter

4) How is 2 Gm written out in English?a) 2 billion metersb) 2 billionths of a meter c) 2 trillion metersd) 2 million meters

5) Which of the following is the longest amount of time?a) 100 million yearsb) 85 Myr c) 14 Gyr d) 3.78 billion years


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Scientific Notation

While metric prefixes provide a way to write large quantities in an abbreviatedway, they are not well suited to do calculations with. We need a short hand for writinglarge numbers that allows for mathematical manipulation. Scientific notation, sometimes

called exponential notation, is this short hand.

The idea behind scientific notation is that when we have a very large or a verysmall number that has been determined by a measurement, we usually dont know all ofthe digits. For example, the distance to the Andromeda galaxy is:

d = 2,900,000 ly

This distance requires a large number to express, but notice we dont know the distance tothe last light year. The only reason to write all of the digits in the number of light years isso we can record the size of the number. The zeros are place holders and in this case only

the left-most two digits are actually known. Scientific notation preserves the knowndigits as a coefficient (in this case 2.9) and then multiplies this by 10 raised to anexponent to account for the place holding zeros. The exponent (sometimes called thepower of ten or simply power) is the same as multiplying10 by itself that manytimes. For example:

d= 2,900,000 ly= 2.9 1,000,000 ly= 2.9 10 10 10 10 10 10 ly= 2.9 106 ly.

The coefficient, the part we actually know from measurement is 2.9 and the exponent,the number of place holders to communicate the size of the number is 6 in 10 6.Another way to say this is we have to shift the decimal point 6 places to the leftto convertthe number to the coefficient.

Now the expression 2.9 106 is not that much of a short hand for 2,900,000. Thefirst expression requires 7 characters to write and the second 9, not much of a difference.But suppose instead of light years we wrote the distance in meters, the standard unit ofdistance in the metric system. In this case the distance to the Andromeda galaxy is:

d= 27,000,000,000,000,000,000,000 m = 2.7 1022 m.

Now it is totally obvious why scientific notation is a good short hand. Imagine trying totype 22 zeros into a calculator and not loosing track as you go.

The same is done for very small numbers. The mass of the electron is:

me = 9.10956 10-31 kg (kilograms).


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Written without scientific notation this is a very cumbersome number:

me = 0.000 000 000 000 000 000 000 000 000 000 910 956 kg

and again the advantage to writing it in scientific notation is immediately obvious. In this

case the coefficient is 9.10956 (we know this mass more precisely, to more digits, thanwe know the distance to the Andromeda galaxy) and the exponent is -31. A negativeexponent is the same as dividingby 10 this many times. Another way to say this is wehave to shift the decimal point 31 places to the rightto convert the number to thecoefficient.

Basic arithmetic rules of manipulation for addition, subtraction, multiplication,and division apply to numbers written in scientific notation. In this class you wont haveto do arithmetic by hand, so discussion will be saved for the next section, ScientificNotation on Your Calculator.

Exercises

A) Write the following numbers in scientific notation.

1) 800000 __________________________________________

2) 0.00000007 __________________________________________

3) 14000000000 __________________________________________

4) 0.0000008374 __________________________________________

5) 670000000000000000 __________________________________________

B) Write the following numbers in normal form.

1) 1.2 102 __________________________________________

2) 3.78 104 __________________________________________

3) 8.5 10-3 __________________________________________

4) 6.49 10-1 __________________________________________

5) 4 100 __________________________________________


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1) The average distance between the Earth and Sun is 93,000,000 miles. This can bewritten _____.a) 9.3 107 miles

b) 9.3 10

-7

milesc) 9.3 106 milesd) 9.3 10-6 miles

2) The wavelength of light the Sun radiates the most of is 0.000 000 55 m. This isequal to _____.a) 5.5 107 mb) 5.5 10 3 mc) 5.5 10-7 md) 5.5 10-12 m

3) Light travels 5.9 trillion miles in one year. This can be written _____.a) 5.9 10-6 miles per yearb) 5.9 10 6 miles per yearc) 5.9 109 miles per yeard) 5.9 1012 miles per year

4) Which of the following numbers is greatest?a) 4.7 10-12

b) 9.8 10 9

c) 0.0 1012

d) 4.1 100

5) Which of the following numbers is smallesta) 4.7 10-12

b) 9.8 10 9

c) 0.0 1012

d) 4.1 100


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Scientific Notation on Your Calculator

Entering Numbers in Scientific Notation into Your Calculator

While the rules for writing and using scientific notation are standardized,

calculator manufactures have not come up with a standard way to either enter or displaynumbers in scientific notation. If your calculator was manufactured in the last five to tenyears it willprobably follow one of the following conventions. If you think it doesntcheck with your instructor and see if together you can figure out how your calculatorworks. But try the following first.

To enter a number in scientific notation you will use a button labeled eitherEXP,

EE, or perhaps EEX. Locate this button on your calculator. Keep in mind it may be asecond function button, that is a button usually labeled in a second color and whichrequires you to push the second function key to use.

To enter a number in scientific notation do notuse any of the buttons labeled, 10x

,ex , xy , yx , or the keys labeled 1 0 ^. If you use any of these you will likely get thewrong answer!

For example, suppose you need to enter the number,

5.87 1013

into your calculator. To do so you press 5 . 8 7 1 3. (The < and > simply indicatethat the button is labeled with the letters in between. Do not look for a < and > button.)

If your calculator does not have a button labeled EE then you likely enter this number by

pressing, 5 . 8 7 1 3. Again do not press the buttons , , , , orthe keys . If you use any of these you will likely get the wronganswer!

Once you have entered this number, or if this number is the result of a calculation,your calculator may display it in a number of possible ways. Unfortunately very fewcalculators will include the 10 part of the number but rather will imply it via one ofthe following conventions. Some possibilities are:

5.8E135.8 13 (a space separating the 8 and the 1)5.8 13

Be sure you understand how your calculator displays scientific notation so you dontincorrectly write down what it is displaying. If for example the above number was theanswer to a question, and on your answer sheet you write 5.8 13, you will get thequestion marked wrong. Mathematically 5.8 13 means to raise 5.8 to the 13th power, notthe same thing as 5.8 1013 power. It is your responsibility to translate correctly whatyour particular calculator displays.


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If instead of a large number like 5.87 1013 the number you are entering is lessthan 1, say 5.87 10-13, you enter it in exactly the same way but you also have to enter thenegative sign that goes along with the exponent. On most calculators the negative sign isnot the same as the minus sign. This may be somewhat counter-intuitive until yourealize that to the calculator the minus sign means to subtract two numbers while the

negative sign means the negative of a number. The minus sign, meaning subtraction, willbe indicated by a button, usually next to, below, or above the addition or plus button,

. The negative button will appear elsewhere and may be marked as () , CHS (for

change sign), or+/. To enter the number 5.87 10-13 press 5 . 8 7 1 3.

Finally, if the number is a negative number with a negative exponent, then youwill need to use the negative key twice. For example, to enter -5.87 10-13 into your

calculator press 5 . 8 7 1 3.

Step by step example:

Suppose you are doing an in class activity and your calculator displays thefollowing number as an answer to a problem,

9800000000.

How do you write this number on the activity sheet? Any number larger than 100,000

or smaller than 0.001 should be written in scientific notation for simplicity and toprevent mistakes in transcription (writing too many or too few zeros). To write thisnumber in scientific notation first determine the coefficient, the known part. In this casethat is the 98 at the beginning, so write this as 9.8 (the coefficient should always have onedigit to the left of the decimal and the rest to the right), then determine the exponent toput with the 10. To do this, simply count how many times you have to move the decimalto take it from where it is in the number to where it is in the coefficient. In this case 9places to the left.4 Then put these together in the scientific notation form:

9.8 109.

If the result from your calculator is a number less than 1, say:

0.00000786,

follow the same procedure except this time you will be shifting the decimal point to the

4 Most scientific calculators have some sort of menu/mode button that will set the calculator into a modeso that it will display allnumbers in scientific notation. If you set your calculator this way you will nothave to count zeros, but you will have to deal with small numbers being displayed in scientific notation.Every make and model of calculator does this differently, so if you want to use this functionality of yourcalculator, you will have to take the initiative and read your calculators manual or otherwise figure outhow to set it on your calculator.


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right. In this case the coefficient is 7.86 and the exponent is -6. So in scientific notationthe number is written as:

7.86 10-6.

Often the number displayed on the calculator does not have place holding zero-digits but rather is just a long string of numbers. What do you do then? You have toconsider where the answer came from, and how well you knew the measurements thatwent into calculating the number displayed. The number of known digits in the resultingnumber is the same as the number of known digits is the leastwell known of the numbersthat went into the calculation. Suppose you calculated the following and got this result:

You should not write 2636231974 as the answer. If you do, you are stating that you know

the answer to 10 digits, but you dont. The top two numbers going into the calculationwere each known to 3 digits, and the bottom number was known to 6 digits. The answeris only known to the same number of digits as the least well known of the numbers goinginto the calculation, so in this case that is 3. The answer should be written in scientificnotation and its coefficient should have 3 digits, 2.64, rounding up the last digit becausethe digit following it is greater than 5. The exponent is then the number of places thedecimal point has to be moved going from the number to the coefficient, in this case thatis 9 places to the left. So the number should be written as:

2.64 109.

Why not just write down what the calculator says? Because the calculator is mindless. Ithas no idea what you are doing. It can only do basic arithmetic. You have to be themindful one and think a bit about what you are trying to work out. This may seem a bitcumbersome at first, but with practice it becomes easy and automatic, and saves you fromwriting out long strings of numbers.

Exercises

A) Type the following numbers into your calculator and hit the equal button, .

1) 4.26 1015

2) 5.31 10-13

3) -8.742 10-18

4) 57,874,000,000,000,000

5) 0.000 000 000 000 345


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B) Do the following calculations on your calculator and write down the result inscientific notation. Be careful to write the correct number of digits in eachcoefficient.

1) = __________________________________________

2) = __________________________________________

3) = __________________________________________

4) = __________________________________________

5) = ____________________________


You will never be asked to demonstrate your skill with scientific notation orarithmetic manipulation on your calculator on an astronomy exam. However, you willneed to know how to use these skills to do many other types of problems.


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Percentage

Percentage is in very common usage in our society, especially in financial issues.Stores will often advertise a 50% off sale. Banks will list loan interest rates, and savingsaccount interest rates as a percentage of the balance per year. Percentage is also often

used in astronomy when referring to a fraction of something. For example, the Earth hashad multi-cellular life for 16% of its existence, or approximately 50% of all solar systemshave more than one star. To be a successful astronomy student or an informed consumerone must know what these percentages actually mean. Fortunately, gaining this skill isone of the more straightforward math skills to acquire.

Percent is an English word which comes from the Latin phrase meaning perhundred or for every one hundred. So, 50% simply means, 50 out of every 100.Sixteen percent, 16%, means 16 per 100. If 16% of the class gets an A, then for every100 students in the class 16 got an A.

The reason 100 is chosen is because for most things we are likely to use a percentfor, 100 is large enough that we will likely get at least one. It is also very easy to use inour base-ten number system, because to change a decimal fraction into a percent, all wehave to do is move the decimal place twice to the right (which of course is the same asmultiplying by 100). Lets illustrate with some examples:

Example 1: You pick up an activity and discover you scored 8 out of the 10 possiblepoints. You want to know what letter grade this is, but you only know how letter gradesequate with percent scores. So what percent is 8 out of 10?

To calculate a percentage when you have 8 out of 10, or any number out of

another, simply calculate the decimal fraction of the first number divided by the second:

8 out of 10 =

then move the decimal point twice to the right or if you prefer multiply by 100. Eitherwill give you the same answer:

8 out of 10 = 0.800 = 80%.

What you are doing when you multiply by 100 is converting the decimal fraction to theout of 100 part of percentage. So, in this case, you scored 80%. Now you can find outwhat letter grade you scored on this assignment.

Example 2: This is the same as example 1, but is a bit more scientific sounding, and ontopic for an astronomy course. There are 128 know stars within 20 light years of theEarth. Of these 128 stars, 86 are M dwarf stars, the coolest type of main sequence starand the most abundant type of star.

What percentage of the stars within 20 ly of the Earth are M dwarf stars?


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Again, to calculate the percentage, first calculate the decimal fraction of thenumber divided by the total. In this case:

86 out of 128 =

then move the decimal point twice to the right or if you prefer multiply by 100, and youget:

67.2% of the stars within 20 light years of the Earth are M dwarf stars.

Now lets take this a step further and do another common type of percentagecalculation. Stars found nearby the Sun, some times called the solar neighborhood, arerepresentative of all of the stars found in the disk of our galaxy. The disk of our galaxycontains approximately 100 billion stars, so based on the percentage we just calculated,how many M dwarfs are there in the disk of our galaxy?

The first thing to note is that stars found near the Sun are representative of all thestars in the disk of the galaxy. Since 67.2% of the stars near the Sun are M dwarfs, 67.2%of all the stars in the disk are also M dwarfs. So the question is asking, how many is67.2% of 100 billion?

The nice thing about percentages is that to calculate the number if we know thepercentage and the total number, all we have to do is multiply the percentage (written inits decimal fraction form) times the total number:

67.2% of 100 billion = 0.672 100 billion = 0.672 (1.00 1011) = 6.72 1010.

So there are 6.72 1010 (67 billion, 200 million) M dwarf stars in the disk of our galaxy.

If this isnt obvious to you, we can work in out by comparison. Recall that 67.2%means 67.2 out of 100 and that this can be written as the fraction:

.

There are 100 billion stars in the disk of the galaxy, so to calculate the number ofM dwarfs (represented byx in the following expression) by comparison we can write:

.

In English, 67.2 out of 100 is the same as what,x, out of 100 billion? Crossmultiplying we find:


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and dividing each side by 100 we get:

.

So the number of M dwarfs in the disk of the galaxy is 6.72 1010, the same number we

calculated previously.

Exercises

A) For each of the following calculate the percentage.

1) 48 out of 100 = ______________________________________________%

2) 6 out of 10 = ______________________________________________%

3) 53 out of 67 = ______________________________________________%

4) 75 out of 115 = ______________________________________________%

5) 345 out of 1258 = ______________________________________________%

B) For each of the following calculate the number.

1) 57% of 100 = ______________________________________________

2) 70% of 10 = ______________________________________________

3) 24% of 28 = ______________________________________________

4) 14% of 221 = ______________________________________________

5) 38% of 2.87 1011 = ______________________________________________


You will never be asked to directly demonstrate your skill with percentage on anastronomy exam. However, you will need to know how to use these skills to do manyother types of problems (like example 2 above), and will need them to understand yourgrades on exams and other assignments.


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The Greek Alphabet

As we will see in the next section, Understanding Equations, we often useletters to represent certain quantities of measure such as distance, mass, etc. Some ofthese are obvious like m for mass, dfor distance. This is a nice convention because it is

easy to remember when looking back at a equation what m and dmean and both m and dare easier to write in an equation than mass ordistance. There are also some standardmathematical conventions which use particular letters for certain purposes like,x andy asgeneral coordinates, i as a counting variable, etc. With only 26 letters in the Romanalphabet, 52 if we count capitals, we rapidly run out of letters. An additional limitation isthat some physical quantities start with the same letter of the alphabet, distance anddensity for example. If we used dfor both, we could easily get confused.

To overcome this limitation we need more letters. We could use any symbols toaugment the standard Roman alphabet. Traditionally many academics (includingundergraduates!) spoke/read Latin and Greek. Latin uses the Roman alphabet, so Greek

letters were the next most familiar in the academic environments and were used for manypurposes. (Think of sororities and fraternities, for example.) So the Greek alphabet waschosen as the next set of symbols to represent many physical quantities.

The following is the upper and lower case Greek alphabet, in alphabetical order.The letters highlighted with a gray background are the ones you are most likely toencounter in introductory astronomy, and so you should learn their names and how towrite them.

A alpha N nu

B beta Xi

gamma O o omicron

delta pi

E epsilon P rho

Z zeta , sigma5

H eta T tau

theta Y upsilon

I iota phi

K kappa X chi

lambda psi

M mu omega

Some Greek letters are similar or identical to letters in the Roman alphabet. This isbecause the alphabet we use is in part derived from the Greek alphabet.

5 There are two forms of lowercase sigma. Written at the end of a word it is written . Written anywhereelse it is written . In introductory astronomy we only use the and forms.


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Exercises

A) Practice printing each of the Greek letters highlighted in gray in the table.Practice until you can print each letter without looking at the table. This way youwill be able to write each in your notes when you need to.

Alternatively, if you take notes on your computer, figure out how to enter theminto your word processor. Some word processors will even let you set upkeyboard macros for special characters, so you may want to define such macrosfor each of the Greek letters highlighted in gray in the table.


You will never be asked to draw or identify a Greek letter on an astronomy exam,however you will need to be able to interpret them in formulas and other situations.


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Understanding Equations

Many students treat mathematical equations (or formulas) as mystical blackboxes, memorized, used to plug and chug, but generally without meaning. Nothingcould be further from the truth (or the best way to use them to get good grades!). Most

equations in physical science convey a fundamental understanding about the physicalworld, the relationship between properties of an object or even properties of the entireUniverse. Thinking about equations in this way is not only a good way to understandthem, but will create understanding about the concepts they are representing. You shouldbe able to plug and chug when you need to, but more fundamentally you should be ableto read an equation and describe, in words, what it says about the physical world.

Example 1: We all understand the relationship between distance, velocity, and time. Ifyou think you dont, consider a road trip. It is about 120 miles from Bowling Green toColumbus, Ohio. It takes about two hours to drive from Bowling Green to Columbus.On average (including stops, turns, etc.) we drive about 60 miles per hour on a highway.

All of these statements are related. It takes two hours to drive to Columbusbecause of the distance Columbus is from Bowling Green and because of how fast wedrive. If you drive faster than and average person, you can get there more quickly. If youdrive slower, it takes more time. If instead of driving to Columbus you drive to a closercity, Toledo say, it will take you less time to get there. If you drive to a further city,Chicago for example, it takes more time to get there.

All of these ideas put together represent a formula relating distance, time, anddriving velocity. We can describe it in words as in the above paragraph, or we can usethe much more compact and precise language of mathematics and write it as the equation:

where drepresents distance, v represents velocity, and trepresents time.

With this equation you can derive all of the above relationships. For example, ifthe distance is fixed, d= 120 miles, then since velocity time will always be equal to thisnumber of miles it becomes clear that the larger the velocity, the smaller the time will beor the smaller the velocity, the larger the time will be. Consider, if average drivingvelocity is 60 miles per hour, the corresponding average trip time is 2 hours,

.

If you could drive faster than this average driving velocity, say 120 miles per hour,obviously you will get there faster. But how much less the time is, is exactly the same ashow much faster you are driving since an increase in velocity will be offset by exactly thesame decrease in time.


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If you drive twice as fast you get there in halfthe time. If you drive halfas fast (30 milesper hour) it takes twice as much time to get there:

.

When you look at a formula, relationships like these are the things you should belooking for.

Constants

Many formulas that you will encounter in your study of astronomy have one ormore constants in them. A constant is a number which never changes. It is always the

same in the equation. Usually the constant is in the equation so that the units of measure(kilograms, meters, seconds, etc.) balance. For example:

In this equation, represents the wavelength (color) of light, and Tthe surfacetemperature of a star. Since wavelength is a size, it is measured in meters, or billionths ofa meter (nm) in the equation, and surface temperature is inKelvins, the temperature scaleused in astronomy. To make the equation an equality, we need the constant,2,900,000 nm K that tells us how the two units relate to each other in this relationship.

Since the constants are often cumbersome to write out, they are often representedby a letter just like the variables are. The equation that expresses the relationshipbetween the physical size of a star, its temperature and how much light it emits is:

whereL is the amount of light the star emits (luminosity),R is the radius of the star, andTis the surface temperature. The 4 and the are numbers, and the is a constant thattells how square meters, Kelvins, and energy (amount of light) are related.

Keeping this is mind, if we look at this equation, we can tell some fundamentalproperties of stars. For example, if we have two stars with the same surface temperature(Thas the same value) then the bigger star (the one with largerR) will produce the mostlight. If two stars are the same size and one is twice as hot as the other, the hotter starproduces 24 = 16 times as much light as the cooler star. If the stars produce the sameamount of light and are the same size, they musthave the same surface temperatures, andso on.


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Exercises

A) Use the equation relating distance, time, and velocity ( ) to answer thefollowing questions:

1) A drive from Bowling Green to the Northwest Suburbs of Chicago is about 300miles. If you drive 60 miles per hour, how long does the drive take? (This is yourchance to plug and chug.)

2) If you drove half as fast as the speed in question 1, how long does the trip take?(Dont plug and chug. Look at the equation and think about it.)

3) If you drove twice as fast as the speed in question 1, how long does the trip take?

4) If instead of Chicago, you drove twice as far, but at twice the speed of question 1,how long would the trip take?

5) In general, if you drive twice as far and want to get there in half the time, howmuch faster do you have to drive?

B) The relationship between the pressure (P), temperature ( T), and density ( ) of

a gas is , where kis a constant.

1) If the density of a gas doesnt change, and the temperature increases, what willhappen to the pressure?

2) If the pressure of a gas is increased by a factor of 4 (made 4 times larger) and thedensity of the gas doubles, what happens to the temperature?

3) If the pressure of a gas is decreased by a factor of 4 but the density doesntchange, what happens to the temperature?

4) If we measure the temperature of a gas, and determine its density fromexperiment, what can we tell about the pressure without having to make anyadditional measurements?

5) If both the density and temperature of a gas are increased by a factor of 10, whathappens to the pressure?


Exam questions will be based on other astronomical equations. Before suchquestions are relevant, we need to first discuss the equations. If you can do the above 10questions you understand how to read and work with equations.


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Answers to Exercises

The following are the answers to the exercises at the end of each section. Only check theanswers once you have finished the exercises. If you look beforehand the exercises willdo very little to help you learn the concepts.

Exercises Names of Numbers

A) Write the following numbers numerically.

1) one million 1,000,000

2) two billion 2,000,000,000

3) five trillion 5,000,000,000,000

4) twenty-three million 23,000,000

5) fifty-seven trillion 57,000,000,000,0006) two hundred thirty-eight billion 238,000,000,000

7) seventy-eight billion,three hundred eighty-eight million

78,388,000,000

8) four hundred fifty-three billion,eight hundred sixty-one million

453,861,000,000

9) five hundred sixty-nine trillion,four hundred seventy-one billion,two hundred thirty-two million

569,471,232,000,000

10) nine hundred fourth-seven trillion,six hundred ninety-two billion,twenty-one million,six hundred one

947,692,021,000,601

B) Write the names of the following numbers in English.

1) 3,000,000 three million

2) 5,000,000,000 five billion

3) 7,000,000,000,000 seven trillion

4) 56,000,000 fifty-six million

5) 73,000,000,000,000 seventy-three trillion

6) 868,000,000,000 eight hundred sixty-eight billion

7) 97,142,000,000 ninety-seven billion, one hundred forty-twomillion


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8) 885,479,000,000 eight hundred eighty-five billion,four hundred seventy-nine million

9) 421,933,593,000,000 four hundred twenty-one trillion,nine hundred thirty-three billion,five hundred ninety-three million

10) 237,807,046,612,739 two hundred thirty-seven trillion,eight hundred seven billion,forty-six million,six hundred twelve thousand,seven hundred thirty nine

Sample Exam Questions Names of Numbers

1) Which of the following is the numerical equivalent of five million?

a) 5,000,000

b) 50,000c) 500,000d) 5,000,000,000,000

2) Which of the following is the numerical equivalent of eight trillion?a) 8,000,000,000,000,000,000,000b) 8,000,000

c) 8,000,000,000,000

d) 800,000,000

3) Which of the following is the is the name for 500,000,000?a) five hundred thousand millionb) five hundred trillionc) five hundred billion

d) five hundred million

4) Which of the following is the name for 434,346,551,000?a) four hundred thirty-four trillion, three hundred forty-six billion, five

hundred fifty-one thousand

b) four hundred thirty-four billion, three hundred forty-six million, five

hundred fifty-one thousand

c) four hundred thirty four kzillion, three hundred forty-six bzillion, fivehundred fifty-one million billlion

d) four hundred thirty-four million, three hundred forty-six thousand, fivehundred fifty-one

5) Of the following four numbers, which is largest?a) 900,342,797

b) one hundred three trillion

c) 141,075d) two hundred ninety-two billion


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3) How is 550 nm written out in English?a) Five hundred fifty million meters.b) Fifty-five hundred billionths of a meter

c) Five hundred fifty billionths of a meter

d) Five hundred fifty trillionths of a meter

4) How is 2 Gm written out in English?

a) 2 billion meters

b) 2 billionths of a meter c) 2 trillion metersd) 2 million meters

5) Which of the following is the longest amount of time?a) 100 million yearsb) 85 Myr

c) 14 Gyr

d) 3.78 billion years


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4) Which of the following numbers is greatest?a) 4.7 10-12

b) 9.8 109

c) 0.0 1012

d) 4.1 100

5) Which of the following numbers is smallest

a) 4.7 10-12

b) 9.8 10 9

c) 0.0 1012

d) 4.1 100


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Exercises Scientific Notation on Your Calculator

A) Type the following numbers into your calculator and hit the equal button, .

1) 4.26 1015 No answers to display. Your answer is correctly

2) 5.31 10-13 getting the number typed into your calculator.

3) -8.742 10-18

4) 57,874,000,000,000,000

5) 0.000 000 000 000 345

B) Do the following calculations on your calculator and write down the result inscientific notation. Be careful to write the correct number of digits in eachcoefficient.

1) = 4.5 1010

2) = 5.71 1016

3) = 2.25 101 = 22.5

4) = 8.72 107

5) = 3.54 1022


You will never be asked to demonstrate your skill with scientific notation orarithmetic manipulation on your calculator on an astronomy exam. However, youwill need to know how to use these skills to do many other types of problems.


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Exercises Percentage

A) For each of the following calculate the percentage.

1) 48 out of 100 = 48%

2) 6 out of 10 = 60%

3) 53 out of 67 = 79%

4) 75 out of 115 = 65%

5) 345 out of 1258 = 27.4%

B) For each of the following calculate the number.

1) 57% of 100 = 57

2) 70% of 10 = 7

3) 24% of 28 = 6.7

4) 14% of 221 = 31

5) 38% of 2.87 1011 = 1.1 1011


You will never be asked to demonstrate your skill with percentage on anastronomy exam. However, you will need to know how to use these skills to do

many other types of problems, and will need them to understand your grades onexams and other assignments.


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Exercises

A) Use the equation relating distance, time, and velocity ( ) to answer thefollowing questions:

1) A drive from Bowling Green to the Northwest Suburbs of Chicago is about 300miles. If you drive 60 miles per hour, how long does the drive take? (This is yourchance to plug and chug.)

If you drive 60 miles per hour the drive takes:

.

The drive takes 5 hours.

2) If you drove half as fast as the speed in question 1, how long does the trip take?(Dont plug and chug. Look at the equation and think about it.)

If you drive have as fast, v is half as much, so to cover the same distance t has tobe twice as much. If you drive half as fast, the trip takes twice as long.

3) If you drove twice as fast as the speed in question 1, how long does the trip take?

By similar reasoning to that of question 2, if you drive twice as fast the trip takeshalf as long.

4) If instead of Chicago, you drove twice as far, but at twice the speed of question 1,how long would the trip take?

In this case d is twice as much, but so is v, so the time is unchanged. This triptakes 5 hours just like the one in question one did.

5) In general, if you drive twice as far and want to get there in half the time, howmuch faster do you have to drive?

If you drive twice as far, d is twice as large, and want to get there in half the time,t is half the value, then you need to drive 4 times as fast.


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B) The relationship between the pressure (P), temperature ( T), and density ( ) of

a gas is , where kis a constant.

1) If the density of a gas doesnt change, and the temperature increases, what willhappen to the pressure?

If we dont change but increase T, then P will increase the same amount as T.

2) If the pressure of a gas is increased by a factor of 4 (made 4 times larger) and thedensity of the gas doubles, what happens to the temperature?

The temperature would also have to double so when combined with the doublingof the density, we get 4 times as much pressure.

3) If the pressure of a gas is decreased by a factor of 4 but the density doesntchange, what happens to the temperature?

If the pressures is one quarter as much and density is unchanged, the temperaturewould also drop to one quarter its value.

4) If we measure the temperature of a gas, and determine its density fromexperiment, what can we tell about the pressure without having to make anyadditional measurements?

We can tell exactly what the pressure is just from our knowing the temperature

and density. These physical properties always follow the relationship,

so we can use it to determine the pressure.

5) If both the density and temperature of a gas are increased by a factor of 10, whathappens to the pressure?

The pressure would increase by a factor of 100, 10 10.


Exam questions will be based on other astronomical equations. Before suchquestions are relevant, we need to first discuss the equations. If you can do theabove 10 questions you understand how to read and work with equations.

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